UNIVERSITY
OF TRENTO
DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE
38123 Povo – Trento (Italy), Via Sommarive 14
http://www.disi.unitn.it
ITERATIVE IMAGE RECONSTRUCTION OF TWO-DIMENSIONAL
SCATTERERS ILLUMINATED BY TE WAVES
D. Franceschini, M Donelli, G. Franceschini, and A. Massa
January 2011
Iterative Image Reconstruction of Two-dimensional
Scatterers Illuminated by TE Waves
Davide Franceschini, Massimo Donelli, Gabriele Franceschini, and Andrea Massa
Abstract
The iterative reconstruction of unknown objects from TE measured scattered field data is presented. The paper investigates the performance of the iterative multiscaling approach (IMSA) in exploiting transverse electric (TE) illuminations. As a matter of fact, in these conditions, the problem turns out to be more complicated than the TM scalar one in terms ofmathematical model as well as computational costs. However,it is expected that more information on the scenario under test can be drawn from scattered data.Therefore, this work is aimed at verifying whether the TE case can provide additional information on the scenario under test (compared to the TM illumination) and how such an enhancement can be suitably exploited by the IMSA for improving the reconstruction accuracy of the retrieval process. Such an analysis will be carried out by means of a set of numerical experiments concerned with dielectric and metallic targets in single and multiple objects configurations. Synthetic as well as experimental data will be dealt with.
Index Terms
Microwave Imaging, Electromagnetic Scattering, Inverse Problems, Multiresolution Technique, TE Illumination.
I. INTRODUCTION
The reconstruction of inaccessible areas has been addressed employing electromagnetic fields at microwave frequencies in
a wide number of applications (see [1][2][3][4]).The imaging ofthe unknown targets is generally achieved through a suitable
processing of the field data collected away from the scatterers. Moreover, the exploitation of the complex scattering phenomena allows a quantitative description of the dielectric and conductivity distributions of the domain under test.
In this framework, two main categories of methodological approaches have been developed. The first class concerns with approximate techniques, such as the physical optic methods [5][6] and the Born and Rytov approximations [7][8]. These approaches simplify the imaging problem by considering some approximations, but also limit the range of retrievable profiles. A second class of methods aims at rigorously solving the nonlinear inverse scattering equations, usually reformulating the original problem into an optimization one [9][10][11] [12][13][14][15]. Such approaches extend the range of retrievable objects but they also make the reconstruction problem more troublesome and computationally expensive since the arising non-linearity and ill-posedness have to be suitably addressed.
Moreover, although many efforts (see for example [16], [17], [18] and [19]) have been addressed to develop effective methods for 3D reconstructions, tomographic configurations are generally considered if possible, since the dimensionality of the problem requires less computational load.
Furthermore, the inverse scattering problem is generally simplified by considering suitable illumination conditions. The most widely adopted polarization is the transverse magnetic (TM) one, where the incident electric field is directed along the invariance
axis of the cylindrical geometry. Such a situation allows to reformulate the original vectorial problem by means of scalar
relations, thus reducing the overall complexity of the mathematical model and the arising computational burden needed for its numerical solution.
On the other hand, dealing with a transverse electric (TE) illumination (where the magnetic field is polarized along the axis of the cylinder) requires a more complex mathematical description since the arising vectorial integral equations present derivatives of both the background Green’s function and the field [20]. Nevertheless, since TM and TE cases are physically uncoupled,
D. Franceschini, M. Donelli, G. Franceschini, and A. Massa are with the Department of Information and Communication Technology, University of Trento, Via Sommarive 14 - 38050 Trento - ITALY, Tel.: +39 0461 882057, Fax: +39 0461 882093, E-mail: andrea.massa@ing.unitn.it, {massimo.donelli,
TE scattered data are expected to give different information on the scenario under test [21] and maybe a larger amount since two field components are taken into account. Moreover, as pointed out in [21], the influence of the ill-posedness for TE
inverse scattering is lower than that for TM since the Green’s function is more singular in the former case. However, the
computational load for exploiting such positive features is unavoidably increased because of the need of solving a vectorial
problem characterized by a stronger non-linearity [22]. As a matter of fact, by considering the same spatial resolution for
reconstructing the unknown profile in the investigation domain, the dimension of the space of unknowns turns out to be larger for TE than for TM case. Thus, retrieval procedures able to effectively deal with large-dimension problems are necessary.
The need of a high resolution accuracy in presence of a limited information content of collectable data [23] hasbeen successfully
addressed for TM data by considering multiresolution strategies [24][25][26][27][28][29], which also allows an increasing of
the ratio between data and unknowns useful for reducing the occurrence of local minima [30]. Therefore, what this paper
accomplishes is to efficiently use a multiresolution strategy [i.e., the Iterative Multi-Scaling Methodology (IMSA)] for TE scattering data and compare the performance to the TM case. This is the first time, to the best of the authors’ knowledge, that a multiresolution methodology is applied to TE scattered data.
The paper is organized as follows. After the mathematical formulation of the inverse scattering problem (Sect. 2), Section 3 will focus on the implementation details of the multistep reconstruction algorithm in dealing with a transverse electric polarization. Successively, the advantages of the IMSA-TE will be analyzed and discussed in Sect. 4 through a numerical analysis concerned withthe reconstruction of dielectric and metallic profiles from both synthetic and experimental data.Finally,some conclusions and future developments will be outlined (Sect. 5).
II. MATHEMATICALFORMULATION
Let us consider a set of monochromatic incident electric fields (Ev
inc(r), v = 1, ..., V ) at a fixed angular frequency ω that
illuminates, from V different angles of incidence, an inhomogeneous object located in a bounded investigation domain DI
lying in free-space (Fig. 1). The unknown object is described through the so-called object function τ (r) given by
τ (r) = εr(r) − 1 − j
σ (r) ωεo
(1) whereεr(r) is the relative permittivity with respect to the background and σ (r) is the conductivity of the scatterer. The total
electric field Ev
tot(r), v = 1, ..., V , inside and outside the scatterer satisfies the following integral equation
Evtot(r) = Ev inc(r) + k20 Z DI τ (r′)Ev tot(r′) · G(r|r′) dr′ (2) wherek2
0= ω2µ0ε0,G(r|r′) is the dyadic Green’s functionand r′ ∈ DI identifies a point inside the investigation area, while
r refers to a location of the investigation domain or of the measurement domain DM external toDI.
Equation (2) is the basic relationship for developing any inversion algorithm based on the integral equation formulation. Moreover, in the following, a tomographic configuration will be assumed [τ (r) = τ (x, y), z being the axis of symmetry of
the scatterer].
For 2D TM polarization, the electric field is parallel to the invariance axis of the cylindrical target, Ev
inc(x, y) = Ez,incv (x, y)ˆz.
Thus, equation (2) reduces to a scalar relation Ev z,tot(x, y) = Ez,incv (x, y) + k20 R DIG(x, y|x ′, y′)τ (x′, y′) Ev z,tot(x′, y′) dx′dy′ ∀(x, y); (x′, y′) ∈ DI (3) where G(x, y|x′, y′) = −j 4H (2) 0 k0 q (x − x′)2+ (y − y′)2
, H0(2) being the second kind 0th-order Hankel function and
j2= −1.
Because of the non linear nature of the problem in hand and its intrinsic ill-posedness, the electromagnetic inversion is
performed by looking for a configuration of the unknowns [i.e., τ (x, y) and Ev
z,tot(x, y) in DI] that minimizes the fitting
For 2D TE polarization, the problem in hand turns out to be more complicated since it cannot be described in terms of
electric-field formulation through a scalar formulation(1)1. In this case,Ev
inc(x, y) = Ex,incv (x, y)ˆx+ Ey,incv (x, y)ˆy.
The domain integral equation for the electric field vectors is given by
Evtot(r) = Evinc(r) + (k20+ ∇∇·)
Z
DI
G(r|r′) · τ (r′)Ev
tot(r′) dr′
where the spatial differentiation operates on r.
Starting from this vectorial equation, a TE-based inversion procedure is aimed at retrieving the object function τ (x, y) and
simultaneously two components of the electric field [i.e., Eυ
x,tot(x, y) and Ey,totυ (x, y), v = 1, ..., V ] in DI.
III. THEMULTI-SCALINGAPPROACH FOR THETE PROBLEM
Starting from the mathematical description in terms of Data and State equations, the inverse problem is solved by means of the iterative multiscaling procedure. Since the IMSA has been widely described for the TM case [26][27], in the following only the main issues concerned with the TE case will be briefly resumed.
In general, the main motivation for adopting a multiresolution methodology lies in the limited information content of the collectable data [23] that does not allow an arbitrary resolution level for the unknowns in the investigation domain. Especially for the TE case, the optimal representation of the unknown has to be efficiently addressed since such a vectorial problem is characterized by more unknown parameters with respect to the TM scalar one.
As a matter of fact, the IMSA defines a suitable discretization of the unknown distributions according to the available data[31].
At each step (p being the step index) of the multiscaling process, only a limited number of N(p) basis functions is employed
for representing the unknowns in the region-of-interest (RoI) where the scatterer is supposed to be located. Iteratively, the same number of basis functions is reallocated in a reduced area defined according to the criteria given in [27]. Therefore,
the original investigation domain (i.e., DI) turns out to be discretized in a non-uniform way since the spatial resolution is
step-by-step enhanced only in a limited sub-domain of DI. Such a zooming strategy limits the total number of unknown
coefficients (weighting the N(p) basis functions) allowing a detailed representation of the scenario where necessary.
As far as the retrieval of the profile under investigation is concerned, the values of the TE unknowns, i.e. τ xn(p), yn(p),
Ev
x,tot xn(p), yn(p), E
v
y,tot xn(p), yn(p), are obtained by minimizing the following cost function
Φ(p)τ xn(p), yn(p) , E v c,tot xn(p), yn(p) ; c = x, y n(p)= 1, ..., N(p) v = 1, ..., V = ΦData(p) + ΦState(p) ΦData (p) = P c=x,y n PV v=1 PM(v) m(v)=1 αData c Ec,totv xm(v), ym(v) − E v c,inc xm(v), ym(v) −PN(p) n(p)=1 P d=x,yτ xn(p), yn(p) E v d,tot xn(p), yn(p) Gcd xm(v), ym(v) xn(p), yn(p) 2o ΦState (p) = P c=x,y n PV v=1 PN(p) n(p)=1βStatec Ec,incv xn(p), yn(p) −hEv c,tot xn(p), yn(p) − P N(p) u(p)=1 P d=x,yτ xu(p), yu(p) Ev d,tot xu(p), yu(p) Gcd xn(p), yn(p) xu(p), yu(p) i 2 (4) where αData c = PV αc v=1 PM(v) m(v)=1 ˛ ˛ ˛E v c,tot “ xm(v),ym(v) ” −Ec,incv “ xm(v),ym(v) ”˛ ˛ ˛ 2 (5) and βcState= βc PV v=1 PN(p) n(p)=1 Ev c,inc xn(p), yn(p) 2. (6)
1 In the TE case, the domain integral equation can be formulated as a scalar-domain integral equation with the one non-zero magnetic-field component as the unknown field or in terms of a vector integral equation with the electric-field vector (two nonzero components) as the unknown field. In [20], it has been shown that the single magnetic-field formulation turns out to be inferior to the dual electric-field formulation.
Moreover,αc andβc are weighting coefficients related to thec-th field component; Gxx,Gxy= Gyx, andGyy are given by Gxx(xh, yh|xk, yk) = −jπakJ1(koak) 2ρ3 hk h koρhky2hkH (2) o (koρhk) + x2 hk− yhk2 H (2) 1 (koρhk) i h 6= k −4jhπkoakH (2) 1 (koak) − 4j i h = k (7) Gxy(xh, yh|xk, yk) = −jπakJ1(koak) 2ρ3 hk xhkyhk h 2H1(2)(koρhk) −koρhkHo(2)(koρhk) i h 6= k 0 h = k (8) Gyy(xh, yh|xk, yk) = −jπakJ1(koak) 2ρ3 hk h koρhkx2hkH (2) o (koρhk) − x2 hk− y2hk H (2) 1 (koρhk) i h 6= k −j4hπkoakH1(2)(koak) − 4j i h = k (9)
where ρhk = px2hk+ y2hk , xhk = (xh− xk) and yhk = (yh− yk). Moreover, J1 is the first order Bessel function and
H1(2) is the second kind Hankel function of first order, respectively;ak =
q
Ak
π ,Ak being the area of the k-th discretization
sub-domain.
The minimization of (4) is iteratively performed with a conjugate-gradient procedure. Even though more efficient optimization techniques are currently under study for the TE case and they have been already developed for the TM case [32], the use of a deterministic technique for this analysis is motivated by the need of focusing on the effectiveness of a “pure” IMSA in dealing with TE date neglecting the ”overboost” effects as well as the randomness arising from its integration with a stochastic optimizer (more effective in avoiding the solution is trapped in the local minima of the cost function).
Finally, the multistep procedure terminates at p = popt when the stability conditions, defined in [27] for the TM case, hold
true.
IV. NUMERICALANALYSIS
In this section, illustrative reconstructions will be displayed and discussed in order to assess the effectiveness of the IMSA in dealing with TE data. Three different scenarios will be considered: the case of a single dielectric scatterer (Sect. 4.1), the case of multiple dielectric scatterers (Sect. 4.2), and the case of metallic targets in both synthetic and experimental environments (Sect. 4.3). The results will consist of gray-level maps of the object functions, variations of the multiresolution cost function versus
the iteration numberk, behaviors of some representative error figures versus the signal-to-noise ratio (SNR) characterizing the
scattered data, and samples of the cost function along fixed directions in the solution space.
A. Single Dielectric Scatterer
The first example deals with a homogeneous square dielectric profile (τ0 = 0.5) of side L0 = 0.48λ0, positioned at
x0 = −0.24λ0, y0 = 0.48λ0 [Fig. 2(a)]. A square investigation domain LDI = 2.4λ0 in side has been assumed and it has
been partitioned in N(p) = 36 cells. The reconstruction has been carried out exploiting multiview data collected from V = 4
different directions of illumination by means of M(v)= 21 receivers located on a circular observation curve of radius 1.8λ0.
The maps of the retrieved profiles during the multistep reconstruction process are reported in Figs. 2(b)-2(e). As it can be
observed, the retrieved object function evolves from a low-resolution representation [τ˜(1)- Fig. 2(b)] to the convergence profile
[τ˜(popt)- Fig. 2(e) and Fig. 3(c)-(d)], achieved after popt= 4 steps and characterized by the lowest values of the error figures
given in Tab. I and computed as follows
δ(s)= r h ˜ x(s)0 − x (s) 0 i2 +hy˜0(s)− y (s) 0 i2 λ0 , s = 1, ..., S (10)
∆(s)= L˜ (s) 0 − L (s) 0 L(s)0 × 100, s = 1, ..., S (11) ξ(s)(i) = popt X p=1 1 N(p)(j) N(p)(j) X n(p)=1 ( ˜ τ(s) x n(p), yn(p) − τ (s) 0 xn(p), yn(p) τ0(s) xn(p), yn(p) ) × 100 (12)
where j ranges over the whole investigation domain (j = int), over the area occupied by the actual scatterers (j = int) or
over the background (j = ext) and the index s indicates the s-th object lying in the investigation domain.
Focusing on such a convergence profile, the numerical values of Tab. I confirm the effectiveness of the inversion algorithm in exploiting TE data for reconstructing the shape of the object (δT E = 2.0 × 10−4,∆T E = 12.33) as well as its dielectric
properties (ξT E
tot = 0.21).
In order to better appreciate the reconstruction accuracy achieved by the IMSA applied to TE data (even though in noiseless conditions), the same profile has been retrieved using TM data. Since the TE illumination provides two components of the field (Ev
x,Eyv) and more information it is expected to be collected on the sensed scenario, an overall imaging enhancement
compared to the reconstruction obtained with a TM illumination should be verified. In order to check such a supposition,
the result of the IMSA-TM inversion is shown in Fig. 3(a)-(b). As expected, despite the reconstruction is satisfactory, the estimated error figures (Tab. I) are higher than those of the TE experiment. In particular, although for a noiseless experiment, the localization accuracy is decreased (δδT MT E ≃ 20) and the overall estimation of the dielectric distribution worse as indicated
by the values of the quantitative error figures: ξ
T M ext ξT E ext ≃ 21.5, ξT Mint ξT E int ≃ 1.4, and ξtotT M ξT E tot ≃ 1.5.
As far as the trade-off between enhancement of the reconstruction accuracy and increasing of the computational costs in dealing with a TE illumination is concerned, Fig. 4 gives some indications on the minimization of the multiresolution cost functionΦ(p)
at the different steps of the IMSA. The spikes occur when the investigation area is scaled and the supports of the basis functions are redefined. In such points, the minimization of the functional starts with an higher level of resolution. For a comparative analysis, the behavior of the cost function for the TM case in shown, as well. As can be noticed, onthe one hand, the TE cost function reaches a lower value and a more accurate fitting with the problem data, thus justifying a better reconstruction. On the other hand, the processing of the TE scattering data requires an additional amount of computational resources (Tab. II) in
terms of both total number of iterations KT E
opt ≃ 4.5KoptT M (Kopt=Pp=1poptk(p)conv,k(p)conv being the number of iterations needed
to achieve “convergence” at the pth step of the multiscaling process) and mean time per iteration (tT E
iter = 120 ms versus
tT M
iter = 36 ms). Such an increment is related to the presence of two components of the fields that contribute at enlarging the amount of independent achievable data as well as the number of unknowns, which causes a larger memory allocation
UT E = 3UT M and processing load.
Because of the increased dimension of the solution space and of the use of a deterministic procedure, it is convenient to analyze
the “shape” of the IMSA cost function as well as the occurrence of local minima in such a functional. As a matter of fact,the
inversion process is performed minimizing (4), thus the complexity as well as the reliability of the reconstruction is strongly related to the presence of local minima that represent false solutions of the problem in hand. For illustrative purposes, Figs. 5(a)-(b) show the plots of the values of the cost function computed in correspondence with an object with the same support of the actual scatterer and varying the value of the object function in the range 0.0 ≤ ˜τ ≤ 16.0. By so doing, a particular
direction of the solution space is explored and some indications about the shape of Φ(p) can be inferred. First of all, let us
observe that, as expected, the global minimum of the cost function turns out to be atτ = τ˜ 0= 0.5.
As can be noticed, the comparison between TE and TM (concerned with V = 1) points out that the number of local minima
for the TE case [Fig. 5(a)] is slightly inferior than for the TM case [Fig. 5(b)]. Moreover, even though the number of views is increased (V ≥ 2), other minimum points in addition to the global one occur in the TM case [Fig. 5(b)], while no local
minima can be observed along this “direction” of the solution space under TE illumination [Fig. 5(a)].
Therefore, although such an example cannot be considered as a definitive indication on the number and on the occurrence of the local minima (since it considers only a sampling direction in the solution space), it seems to confirm also for the TE illumination a key-feature of the IMSA, that is the “reduction” of local minima and, in the first resort/approximation, a certain reason of using a deterministic method as minimizer.
Finally, since the TE exploits the information from two independent field polarizations (Ev
x-T Ex,Eyv-T Ey) and it outperforms
the TM inversion of dielectric profiles, which considers only a scalar field component (Ev
z -T M ), it could be interesting to
assess the effectiveness of the IMSA procedure when each TE scalar component is individually processed. Towards this end, the
inversion process has been carried out by assuming αx= βx= 1.0 and αy = βy= 0.0 (T Exillumination) andαx= βx= 0.0
andαy= βy= 1.0 (T Ey illumination) in the cost function (4). The grey-level maps of the retrieved profilesare shown in Fig.
3(e) and Fig. 3(f ), respectively. The obtained results highlight that only the simultaneous processing of both the TE components is needed for obtaining a more faithful reconstruction than the TM inversion. Moreover, it seems to further confirm that TM and TE illuminations give unrelated data about the scatterer [21] (as a matter of fact different reconstructions have been obtained with the two different datasets), but also that each TE component supplies a differentinformation on the scenario under test.
B. Multiple Dielectric Scatterers
The second example of the numerical validation considers a more complex scenario where a multiple scatterers configuration
lies. Let us refer to the reference geometry shown in Fig. 6 where three equal square cylinders (L(s)0 = 0.67λ0,s = 1, ..., 3,
in side) located at (x(1)0 = y (1) 0 = 0.67λ0), (x(2)0 = −y (2) 0 = −0.67λ0), and (x(3)0 = 0, y (3) 0 = −0.67λ0) are embedded in a
search areaLDI = 4λ0-sided. The value of the object function in the region occupied by the scatterers is equal toτ
(s)
0 = 0.5,
s = 1, ..., 3. The objects have been probed from V = 8 directions and the scattered fields have been collected at M(v)= 35,
v = 1, ..., V positions. Moreover, at the first step (p = 1) of the multiscaling process, the investigation domain has been
partitioned in N(1) = 100 subdomains. For comparison purposes, the reconstruction has been carried out with both TE and
TM illuminations. Moreover, for a more exhaustive assessment also noisy data have been taken into account. Towards this end, the scattered field values have been blurred as in [26] by adding a gaussian contribution with a zero mean value and identified
by a fixedSN R value.
Figure 7 shows in a comparative fashion the achieved results. Pictorially, one can notice a better estimation of the supports of the scatterers when a TE illumination is used. Such an indication is quantitatively confirmed by the values of the error
index ∆ reported in Tab. III and IV. More in detail, when SN R ≤ 10 dB, 11.42 6 ∆(s)T E 616.44 for the TE case, while
its range of variations turns out to be higher in correspondence with TM reconstructions (15.80 6 ∆(s)T M 625.82). As far as
the quantitative imaging is concerned, the total errorξT E
tot is lower than 2.6 in the TE configuration whatever the SN R value,
while ξT M tot SN R=10 dB≃ 3 and ξ T M tot
SN R=5 dB≃ 4.2. Moreover, the values of ξ (s)
int clearly indicate a more accurate retrieval
of the object function in the regions occupied by the scatterers.
For completeness, Tab. V provides some details of the computational burden of the iterative reconstruction processes. When the noise level is negligible, the same conclusions drawn for the single scatterer scenarios hold true and the IM SA − T M
turns out to be more effective in reaching the convergence. On the other hand, the convergence rates are quite similar for lower
SN Rs. However, the IM SA − T E seems to converge faster in strong noise conditions (5dB) and to enhance the accuracy
(ξ
T M
tot|SN R=5 dB
ξT E
tot|SN R=5 dB
≃ 1.7). This points out the positive effect of the exploitation of the two field components especially in critical
measurement conditions.
To further assess the proposed approach with multiple scatterers and noisy conditions, also evaluating the “discrimination”
capabilities of the method, a third experiment concerned with a two-objects configuration [Fig. 8(a) -τ0(s)= 0.5, s = 1, 2] has
been carried out. The scatterers differ in dimensions, L(1)0 = 1.33λ0 andL(2)0 = 0.67λ0, while the illumination/measurement
setup is the same of the previous example.
In order to give a complete overview of the obtained results, Fig. 9 shows the behaviors of the error indexes versus the SN R.
In general, whatever the noise level, the IM SA − T E yields better inversions both in locating [Fig. 9(a)] and shaping [Fig.
approach significantly overcomes the T M -based technique when the noise level grows (i.e., SN R < 15 dB). In such a
situation, the inversion turns out to be more difficult and theIM SA benefits of the enhanced amount of information. Such an
improvement can be also pictorially appreciated in Fig. 8 where the grey-levels maps of the dielectric profiles retrieved when
SN R = 5 dB under T E illumination [Fig. 8(b)] and T M illumination [Fig. 8(c)] are reported, respectively. In particular, for
such a configuration, the details of the error figures are reported in Tab. VI.
As far as the computational burden is concerned, the same conclusions of the previous example can be inferred also for this test case (Tab. VII).
C. Metallic Targets
Finally, the last set of experiments is aimed at evaluating the reconstruction of metallic targets both considering synthetic and
real data. Figure 10(a) shows the reference distribution of the metallic (σ = 20 S/m) object under test of side L0= 0.48λ0
and centered at x0= −0.24λ0,y0= 0.48λ0. As far as the configuration of the experimental setup is concerned, the following
parameters have been used:LDI = 1.6λ0,V = 4, and M(v)= 36. Moreover, DIhas been partitioned inN(1)= 21 subdomains.
The reconstruction results give different indications on the effectiveness of IM SA − T E vs. IM SA − T M in comparison
with those carried out for the dielectric configurations (Sect. 4.1 and Sect. 4.2). As a matter of fact, the profile under test is
generally better imaged when aT M illumination is considered as pointed out by the maps in Figs. 10(b)-(c) (SN R = 10 dB).
Quantitatively,ξT E
tot = 7.16 vs. ξT Mtot = 3.86, ξintT E = 41.72 vs. ξT Mint = 27.73, and ∆T E = 2.2∆T M.
However, such a behavior is not surprising since the same difficulties in dealing with a T E illumination of impenetrable
obstacles have been encountered also by Ramananjaona et al. [33] both using synthetic and real data.
To further analyze such an issue, also laboratory-controlled data concerned with the experimental dataset “Marseille” [34] have
been used. In particular, for a comparative study, the scattered data related to a rectangular metallic target (25.4 × 12.7 mm2)
probed with aT M (“rectTM_cent.exp”) and a T E (“rectTE_8f.exp”) incident field have been considered. Because of the limited
aspect of the measurement, the whole set of data (V = 36, M(v)= 49) at f = 4 GHz has been precessed imposing a constraint
on the real part of object function [Re (˜τ ) = 0] and performing a thresholding (as suggested in [35]) to Υmax = −10.5 if
Im {τ (x, y)} > Υmax.
The reconstruction exploitingT M scattered data [Fig. 11(b)] is more accurate. The boundaries of the object are better detected
and the metallic nature of the scatterer is more faithful estimated. Similar observations have been also made in [33] and [36] where different inversion methods were used. For a preliminary comparison between IMSA-TE and another inversion approach, Fig. 11(c) reports the reconstructed profile obtained with the CSI-TE algorithm [36]. The inversion of the experimental data has been carried out considering single frequency measurements (f = 4 GHz) for reconstructing an investigation region 17 cm
in side. Such single-step strategy achieves a reconstruction in which the dimension of the profile is slightly underestimated, as well as the retrieved value of the imaginary part is greater thanIm {τ (x, y)}minCSI−T E= −0.6 (while Im {τ (x, y)}minIM SA−T E = −3.1). However, for further comparisons with CSI and other different kinds of inversion algorithms the reader may refer to
the papers in [34].
The worse reconstructions with T E data could be related to the measurement setup. As a matter of fact, for the T E case only
one electric field component (i.e., perpendicular to the radial direction) has been measured instead of two orthogonal in the azimuthal plane as required by the vectorial nature of the problem inside DI. Such supposition agrees with other works in
literature on the same dataset, but should be further verified other impenetrable objects or experiments in which the electric field is completely measured.
Moreover, problems are expected with the metallic or high contrast objects when the TE problem is formulated in term of the electric field. As a matter of fact, it is the electric flux rather that the electric field that is continuous in an inhomogeneous medium, right at the boundary of an object. Unfortunately, enforcing this physics in the IMSA would make the gradient quite nonlinear and involved.
V. CONCLUSION
An iterative multiresolution method for the reconstruction of unknown scatterers illuminated byT E incident fields has been
adaptively determined by means of the information acquired during the iterative minimization of a suitable multiresolution cost function.
The numerical analysis carried out through the paper has shown that the proposed methodology presents very attractive
capabilities in imaging single and multiple scatterers and a better reconstruction accuracy compared to the T M illumination
when dielectric scatterers are dealt with. An opposite conclusion has been carried out when metallic scatterers are probed both considering synthetic and real data. This behavior, even though a more exhaustive analysis would be necessary, suggests the
need of extending theIM SA procedure to a hybrid strategy where T M and T E illuminations (when available) are alternatively
processed as proposed in [21].
Moreover, the IMSA-TE has been compared with a single-step CSI-TE inversion algorithm. The multiresolution reconstruction has given better results in estimating the metallic characteristics of the scatterer, however a wider analysis should be carried out in order to fully asses the comparative analysis between the two methodologies. Certainly, it would be interesting to develop a multiscaling procedure based on the contrast source formulation.
Finally, it should be pointed out that the proposed solution of the T E case is an intermediate and mandatory step towards
the solution of the full three-dimensional problem where all the three scalar components of the electric field play their role.
The main differences for such an extension lie in the computation time (expected to substantially increase) as well as in the minimization procedure to be carefully implemented in order to suitably deal with the larger solution space.
ACKNOWLEDGMENT
The authors wish to thank Dr. A. Abubakar for kindly providing the experimental results of the CSI algorithm.
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FIGURECAPTION
• Figure 1. Problem Geometry.
• Figure 2. Reconstruction of an off-centered square dielectric cylinder (τ0 = 0.5) under TE illumination (Noiseless
Conditions). Actual profile (a) and reconstructed profile at: (b) p = 1, (c) p = 2, (d) p = 3, and (e) p = Popt= 4.
• Figure 3.Reconstruction of an off-centered square dielectric cylinder (τ0= 0.5 - Noiseless Conditions). Results obtained
(a),(b) under T M illumination, (c),(d) under T E illumination, (e) exploiting the T Ex component, and (f ) the T Ey
component.
• Figure 4. Reconstruction of an off-centered square dielectric cylinder (τ0= 0.5 - Noiseless Conditions). Behavior of the
multiresolution cost function at the different steps of the minimization procedure for the T E and T M cases.
• Figure 5. Reconstruction of an off-centered square dielectric cylinder (τ0= 0.5 - Noiseless Conditions). Behavior of the
cost functionΦ(p) along a direction of the solution space obtained varying the object function in the range0 < τ < 16:
(a) TE case and (b) and TM case.
• Figure 6. Reconstruction of three square dielectric scatterers (τ0(s)= 0.5, s = 1, ..., 3). Actual profile.
• Figure 7. Reconstruction of three square dielectric scatterers (τ0(s) = 0.5, s = 1, ..., 3). Retrieved profiles under T E
illumination (a)(c)(e) andT M illumination (b)(d)( f ) in (a)(b) noiseless condition, when (c)(d) SN R = 10 dB, and (e)(f )
when SN R = 5 dB.
• Figure 8. Multiple scatterers configuration (SN R = 5 dB). Actual profile (a) and reconstructions achieved under T E
illumination (b) andT M illumination (c).
• Figure 9. Multiple scatterers configuration. Reconstruction errors versus SN R: (a) location errors δ(s), (b) dimensional
errors ∆(s), (c) internal errorsξ(s)
int, and (d) external errors ξ
(s)
• Figure 10. Reconstruction of an off-centered square metallic cylinder (τ0= j89.73). Map of the imaginary part of the
reference object function (a). Reconstruction of the imaginary part of the object function under (b)T E illumination and
(c)T M illumination (SN R = 10 dB) .
• Figure 11. Experimental validation against the real data (Dataset “Marseille” [34]). Reconstruction of the rectangular
metallic cylinder at the frequencyf = 4GHz when probed with (a) a T E incident field ( “rectTE_8f.exp”) and (b) a T M incident field ( “rectTM_cent.exp”). (c) Result obtained with the single-step CSI-TE algorithm.
TABLECAPTION
• Table I. Reconstruction of an off-centered Square Dielectric Cylinder (τ (x, y) = 0.5 + j0.0). Comparison of the error
figures for the TM and the TE illumination.
• Table II. Reconstruction of an off-centered Square Dielectric Cylinder (τ (x, y) = 0.5 + j0.0). Computational Burden. • Table III. Three scatterer configuration. Error figures for the TE illum ination.
• Table IV. Three scatterer configuration. Error figures for the TM illum ination.
• Table V. Three scatterer configuration. Convergence step and total iterations number under TE illumination and TM
illumination.
• Table VI. Multiple Scatterer configuration. Error figures for the TE and TM illumination (SN R = 5dB).
• Table VII. Multiple Scatterer configuration. Convergence step and total iterations number in the TE case and in the TM
scatt
E
v
(x ,y )
m m
z^
x
^
y
^
x
y
θ
vInvestigation Domain
Observation Domain
v
z,inc
E
2D−TM
2D−TE
E
x,inc
v
+E
v
y,inc
ε
cil(x,y)
y x -1.2 -0.8 -0.4 0 0.4 0.8 1.2 x/λ0 -1.2-0.8 -0.40 0.40.8 1.2 y/λ0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 |Re[τ(x,y)]| 0.65 Re{τ (x, y)} 0.0 (a) y x y x
0.65 Re{τ (x, y)} 0.0 0.65 Re{τ (x, y)} 0.0
(b) (c)
y
x
y
x
0.65 Re{τ (x, y)} 0.0 0.65 Re{τ (x, y)} 0.0
(d) (e)
y x -1.2 -0.8 -0.4 0 0.4 0.8 1.2 x/λ0 -1.2-0.8 -0.40 0.40.8 1.2 y/λ0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 |Re[τ(x,y)]| 0.65 Re{τ (x, y)} 0.0 (a) (b) y x -1.2 -0.8 -0.4 0 0.4 0.8 1.2 x/λ0 -1.2-0.8 -0.40 0.40.8 1.2 y/λ0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 |Re[τ(x,y)]| 0.65 Re{τ (x, y)} 0.0 (c) (d) y x y x
0.65 Re{τ (x, y)} 0.0 0.65 Re{τ (x, y)} 0.0
(e) (f )
0.01 0.1 1 10 0 1000 2000 3000 4000 5000 6000 7000 8000 φ (k) Iteration number (k) s=4 s=3 s=2 s=1 s=1 s=2 TE TM
0 10 20 30 40 50 60 70 80 90 0 2 4 6 8 10 12 14 16 Φ Re[τ] v=1 v=2 v=4 v=8 v=16 (a) 0 5 10 15 20 25 0 2 4 6 8 10 12 14 16 Φ Re[τ] v=1 v=2 v=4 v=8 v=16 (b)
y
x
0.65 Re{τ (x, y)} 0.0
y
x
y
x
0.65 Re{τ (x, y)} 0.0 0.65 Re{τ (x, y)} 0.0
(a) (b)
y
x
y
x
0.65 Re{τ (x, y)} 0.0 0.65 Re{τ (x, y)} 0.0
(c) (d)
y
x
y
x
0.65 Re{τ (x, y)} 0.0 0.65 Re{τ (x, y)} 0.0
(e) (f )
y x 0.65 Re{τ (x, y)} 0.0 (a) y x y x
0.65 Re{τ (x, y)} 0.0 0.65 Re{τ (x, y)} 0.0
(b) (c)
0 0.2 0.4 0.6 0.8 1 5 15 25 35 δ SNR [dB] TM-IMSA strategy, δ(1) TE-IMSA strategy, δ(1) TM-IMSA strategy, δ(2) TE-IMSA strategy, δ(2) (a) 0 20 40 60 5 15 25 35 ∆ SNR [dB] TM-IMSA strategy, ∆(1) TE-IMSA strategy, ∆(1) TM-IMSA strategy, ∆(2) TE-IMSA strategy, ∆(2) (b)
0 10 20 30 5 15 25 35 ξint SNR [dB] TM-IMSA strategy, ξ(1) TE-IMSA strategy, ξ(1) TM-IMSA strategy, ξ(2) TE-IMSA strategy, ξ(2) (c) 0 2 4 6 5 15 25 35 ξext SNR [dB] TM-IMSA strategy TE-IMSA strategy (d)
y x 0.0 Im{τ (x, y)} − 40.0 (a) y x 0.0 Im{τ (x, y)} − 40.0 (b) y x 0.0 Im{τ (x, y)} − 40.0 (c)
y
x
y
x
0.0 Im{τ (x, y)} − 10.5 0.0 Im{τ (x, y)} − 10.5
(a) (b)
y
x
0.0 Im{τ (x, y)} − 0.6
(c)
2 3
